Using An Inverse Function To Derive The Rule of 70
The Rule of 70 states: If you invest a given amount of money
at R% interest, it will take approximately 70/R years for your money
to double. Likewise, if the inflation rate is R%, prices will double
in approximately 70/R years. So for example, if you invest
$10,000 at R=7% interest, it will take about 70/7 = 10 years for your
money to double. In another example, if the inflation rate is R=3.5%,
prices will double every 70/3.5 = 20 years.
Where Does This Rule Come From? How To Derive The Rule of 70
The equation governing the growth of money is A =
P●e^{rt} where
r is the interest rate, given as a decimal,
P is the original amount of money,
t is equal to the number of years the money is invested,
e is the exponential base approximately equal to 2.718, and
A is the total amount of money after t years.
If the amount of money P is doubled, the we let A = 2P and the
equation we get is
2P = P●e^{rt},
which simplifies to
2 = e^{rt}
after dividing both sides by P.
Now, we solve this equation for t in terms of r. Here is
where we need the inverse function!
The inverse of the exponential function f(x) = e^{x} is the
natural log function g(x) = LN(x) where
(f o g)(x) = e ^{LN(x)} = x and (g o f)(x) = LN(e^{x})
= x.
So if we take the natural log function of both sides of
2 = e^{rt}, we
get
LN(2) = LN(e^{rt})
which simplifies to
LN(2) = rt
Since LN(2) = approximately 0.693, we can write 0.693 = rt
and solving for t results in
0.693/r = t where r is the rate as a decimal. If we
multiply the left side by 100/100, we get
69.3/(100r) = t. If we let R = 100r where R is the rate
given as a percent, we get
69.3/R = t. To account for less than continuous compounding
and to make a nicer number in our formula, we round the 69.3 up to 70
and our formula becomes:
where R is the rate as a percent and t is the number of years for the
money (or price) to double
